You can put bar-end shifters at the ends of the drop bars! Lots of us do this on our touring/commuting bikes. Here's a
photo of the bar-end shifters on my bike.
But the combination of brakes on the end of the aero bars and shifters on the drop bars would probably be a very bad combination, since you'd have the move your body position greatly in order to move between brakes and shifters! Aerobars are NOT designed for "tooling around the city" kind of use. They're very comfortable for long flat stretches where you don't have to shift or stop and start too much, but they're not any good in traffic, where you need more balance and the ability move between brakes and shifters frequently.